Few remarks on essential modules

TL;DR

This paper classifies modules over integral domains, introduces SM modules, and explores their properties, including relationships with Artinian, locally finite, and finite length modules, as well as socle characteristics.

Contribution

It introduces the new class of SM modules and analyzes their properties and relationships with existing module classes over integral domains.

Findings

SM modules include Artinian, locally finite, and finite length modules as subclasses

Non-semisimple SM modules have essential socles

Modules over integral domains with torsion-free parts lack essential socles

Abstract

In the present paper, modules over integral domains and principal ideal domains that are proper essential extensions of some submodules are classified. We introduce a new class of modules that we call $\mathrm{SM}$ modules and show that the class of Artinian modules, locally finite modules, and modules of finite lengths are all proper subclasses of SM modules. We also show that non-semisimple $\mathrm{SM}$ modules possess essential socles. Further, we show that non-semieimple modules over integral domains with nonempty torsion-free parts do not possess essential socles.